Raising the bar Correlation coefficient: the last block of statistical foundation

Property 1. Range of the correlation coefficient: for any $X,Y$ one has $- 1 \le \rho (X,Y) \le 1$ .
This follows from the Cauchy-Schwarz inequality, as explained here.

Recall from this post that correlation is cosine of the angle between $X-EX$ and $Y-EY$ .
Property 2. Interpretation of extreme cases. (Part 1) If $\rho (X,Y) = 1$ , then $Y = aX + b$ with $a > 0.$

(Part 2) If $\rho (X,Y) = - 1$ , then $Y = aX + b$ with $a < 0$ .

Proof. (Part 1) $\rho (X,Y) = 1$ implies
(1) $Cov (X,Y) = \sigma (X)\sigma (Y)$
which, in turn, implies that $Y$ is a linear function of $X$ : $Y = aX + b$ (this is the second part of the Cauchy-Schwarz inequality). Further, we can establish the sign of the number $a$ . By the properties of variance and covariance
$Cov(X,Y)=Cov(X,aX+b)=aCov(X,X)+Cov(X,b)=aVar(X)$ ,

$\sigma (Y)=\sigma(aX + b)=\sigma (aX)=|a|\sigma (X)$ .
Plugging this in Eq. (1) we get $aVar(X) = |a|\sigma^2(X)$ and see that $a$ is positive.

The proof of Part 2 is left as an exercise.

Property 3. Suppose we want to measure correlation between weight $W$ and height $H$ of people. The measurements are either in kilos and centimeters ${W_k},{H_c}$ or in pounds and feet ${W_p},{H_f}$ . The correlation coefficient is unit-free in the sense that it does not depend on the units used: $\rho (W_k,H_c)=\rho (W_p,H_f)$ . Mathematically speaking, correlation is homogeneous of degree $0$ in both arguments.
Proof. One measurement is proportional to another, $W_k=aW_p,\ H_c=bH_f$ with some positive constants $a,b$ . By homogeneity
$\rho (W_k,H_c)=\frac{Cov(W_k,H_c)}{\sigma(W_k)\sigma(H_c)}=\frac{Cov(aW_p,bH_f)}{\sigma(aW_p)\sigma(bH_f)}=\frac{abCov(W_p,H_f)}{ab\sigma(W_p)\sigma (H_f)}=\rho (W_p,H_f).$